Ratio coordinates for higher Teichm\"uller spaces

TL;DR

This paper introduces new ratio coordinates for higher Teichmüller spaces related to surface representations, generalizes Kashaev's coordinates, and explores their quantization, potentially leading to new representations of mapping class groups.

Contribution

It defines novel ratio coordinates for higher Teichmüller spaces, extends Kashaev's coordinates to higher rank groups, and establishes their quantization for certain cases.

Findings

Defined new ratio coordinates for higher Teichmüller spaces.

Proved consistency of quantization for the case m=3.

Provided a full proof of Kashaev's groupoid presentation.

Abstract

We define new coordinates for Fock-Goncharov's higher Teichm\"uller spaces for a surface with holes, which are the moduli spaces of representations of the fundamental group into a reductive Lie group $G$. Some additional data on the boundary leads to two closely related moduli spaces, the $\mathscr{X}$-space and the $\mathscr{A}$-space, forming a cluster ensemble. Fock and Goncharov gave nice descriptions of the coordinates of these spaces in the cases of $G = PGL_m$ and $G=SL_m$, together with Poisson structures. We consider new coordinates for higher Teichm\"uller spaces given as ratios of the coordinates of the $\mathscr{A}$-space for $G=SL_m$, which are generalizations of Kashaev's ratio coordinates in the case $m=2$. Using Kashaev's quantization for $m=2$, we suggest a quantization of the system of these new ratio coordinates, which may lead to a new family of projective representations of mapping class groups. These ratio coordinates depend on the choice of an ideal triangulation decorated with a distinguished corner at each triangle, and the key point of the quantization is to guarantee certain consistency under a change of such choices. We prove this consistency for $m=3$, and for completeness we also give a full proof of the presentation of Kashaev's groupoid of decorated ideal triangulations.